Question

Express the HCF of $ 468 $ and $ 222 $ as $ 468x + 222y $ to find $ x $ and $ y $ ,where $ x,y $ are integers in two different ways.

Answer 1

Hint: HCF is the largest number that divides two or more numbers as the highest common factor (HCF) for those numbers. For example consider the number $ 30(2 \times 3 \times 5),36(2 \times 2 \times 3 \times 3),42(2 \times 3 \times 7),45(3 \times 3 \times 5) $ are the largest numbers and hence, is the HCF for these numbers.

Complete step-by-step answer:
First find the HCF of $ 468 $ and $ 222 $
Using, Euclid’s division algorithms,
Step 1) $ 222 \times 2 + 24 = 468 $
Step 2) $ 24 \times 9 + 6 = 222 $
Step 3) $ 6 \times 4 + 0 = 24 $
$ \Rightarrow $ HCF $ = 6 $
Now, from step (2), we have
 $ 6 = 222 - 24 \times 9 $
 $\Rightarrow 6 = 222 - (468 - 222 \times 2) \times 9 $ (from step (1))
 $\Rightarrow 6 = 222 - 9 \times 468 + 222 \times 18 $
 $\Rightarrow 6 = 222 \times 19 - 9 \times 468$
 $\Rightarrow 6 = 222 \times ( - 9) + 222 \times 19 $ . . . . (1)
HCF $ = 468x + 222y $ . . . . (2)
From the equation (1) and (2)
 $ x = - 9 $ and $ y = 19 $
Here, we have written
 $ 6 = xa + yb $
 $
 \Rightarrow 6 = xa + yb + ab - ab \\
\Rightarrow 6 = a(x + b) + b(y - a) \\
\Rightarrow 6 = 468( - 9 + 222) + 222(19 - 468) \\
 \Rightarrow 6 = 468 \times (213 )+ ( - 449) \times 222 \;
 $
 $ \Rightarrow x = 213 $ and $ y = - 449 $
Thus, we can express the HCF in different ways.

Note: The highest common factor is found by multiplying all the factors which appear in both lists in mathematics with a number or algebraic expression that divides another or expression evenly. The greatest common measure and greatest common divisor are the other terms used to refer to HCF. We can find out HCF using the prime factorization method or by dividing the numbers or division method.